Gaps and approximations in the space of growth functions

An important problem in combinatorial noncommutative algebra is to give an asymptotic characterization the growth functions of finitely generated algebras (equivalently, semigroups, or hereditary languages). The function every generated, infinite-dimensional increasing and submultiplicative. question what extent these natural necessary conditions are also sufficient—and particular, whether they sufficient at least for sufficiently rapid functions—was posed studied by various authors has attracted a flurry research. While submultiplicative realizable as up linear error term, we show that there exist arbitrarily which not equivalent any algebra, thus resolving aforementioned settling Zelmanov (and repeated Alahmadi–Alsulami–Jain–Zelmanov). These can be interpreted ‘holes’ space functions, accumulating exponential order topology. We monomial languages whose encode existence non-prolongable words, nilpotent ideals (in graded case). This negatively solves another conjecture Alahmadi–Alsulami–Jain–Zelmanov case.